I'm writing a program for a competition and I need to be faster than all the other competitors. For this I need a little algorithm help; ideally I'd be using the fastest algorithm.
For this problem I am given 2 things. The first is a list of tuples, each of which contains exactly two elements (strings), each of which represents an item. The second is an integer, which indicates how many unique items there are in total. For example:
# of items = 3
[("ball","chair"),("ball","box"),("box","chair"),("chair","box")]
The same tuples can be repeated/ they are not necessarily unique.) My program is supposed to figure out the maximum number of tuples that can "agree" when the items are sorted into two groups. This means that if all the items are broken into two ideal groups, group 1 and group 2, what are the maximum number of tuples that can have their first item in group 1 and their second item in group 2.
For example, the answer to my earlier example would be 2, with "ball" in group 1 and "chair" and "box" in group 2, satisfying the first two tuples. I do not necessarily need know what items go in which group, I just need to know what the maximum number of satisfied tuples could be.
At the moment I'm trying a recursive approach, but its running on (n^2), far too inefficient in my opinion. Does anyone have a method that could produce a faster algorithm?
Thanks!!!!!!!!!!
Speed up approaches for your task:
1. Use integers
Convert the strings to integers (store the strings in an array and use the position for the tupples.
String[] words = {"ball", "chair", "box"};
In tuppls ball now has number 0 (pos 0 in array) , chair 1, box 2.
comparing ints is faster than Strings.
2. Avoid recursion
Recursion is slow, due the recursion overhead.
For example look at binarys search algorithm in a recursive implementatiion, then look how java implements binSearch() (with a while loop and iteration)
Recursion is helpfull if problems are so complex that a non recursive implementation is to complex for a human brain.
An iterataion is faster, but not in the case when you mimick recursive calls by implementing your own stack.
However you can start implementing using a recursiove algorithm, once it works and it is a suited algo, then try to convert to a non recursive implementation
3. if possible avoid objects
if you want the fastest, the now it becomes ugly!
A tuppel array can either be stored in as array of class Point(x,y) or probably faster,
as array of int:
Example:
(1,2), (2,3), (3,4) can be stored as array: (1,2,2,3,3,4)
This needs much less memory because an object needs at least 12 bytes (in java).
Less memory becomes faster, when the array are really big, then your structure will hopefully fits in the processor cache, while the objects array does not.
4. Programming language
In C it will be faster than in Java.
Maximum cut is a special case of your problem, so I doubt you have a quadratic algorithm for it. (Maximum cut is NP-complete and it corresponds to the case where every tuple (A,B) also appears in reverse as (B,A) the same number of times.)
The best strategy for you to try here is "branch and bound." It's a variant of the straightforward recursive search you've probably already coded up. You keep track of the value of the best solution you've found so far. In each recursive call, you check whether it's even possible to beat the best known solution with the choices you've fixed so far.
One thing that may help (or may hurt) is to "probe": for each as-yet-unfixed item, see if putting that item on one of the two sides leads only to suboptimal solutions; if so, you know that item needs to be on the other side.
Another useful trick is to recurse on items that appear frequently both as the first element and as the second element of your tuples.
You should pay particular attention to the "bound" step --- finding an upper bound on the best possible solution given the choices you've fixed.
Related
This problem is 4-11 of Skiena. The solution to finding majority elements - repeated more than half times is majority algorithm. Can we use this to find all numbers repeated n/4 times?
Misra and Gries describe a couple approaches. I don't entirely understand their paper, but a key idea is to use a bag.
Boyer and Moore's original majority algorithm paper has a lot of incomprehensible proofs and discussion of formal verification of FORTRAN code, but it has a very good start of an explanation of how the majority algorithm works. The key concept starts with the idea that if the majority of the elements are A and you remove, one at a time, a copy of A and a copy of something else, then in the end you will have only copies of A. Next, it should be clear that removing two different items, neither of which is A, can only increase the majority that A holds. Therefore it's safe to remove any pair of items, as long as they're different. This idea can then be made concrete. Take the first item out of the list and stick it in a box. Take the next item out and stick it in the box. If they're the same, let them both sit there. If the new one is different, throw it away, along with an item from the box. Repeat until all items are either in the box or in the trash. Since the box is only allowed to have one kind of item at a time, it can be represented very efficiently as a pair (item type, count).
The generalization to find all items that may occur more than n/k times is simple, but explaining why it works is a little harder. The basic idea is that we can find and destroy groups of k distinct elements without changing anything. Why? If w > n/k then w-1 > (n-k)/k. That is, if we take away one of the popular elements, and we also take away k-1 other elements, then the popular element remains popular!
Implementation: instead of only allowing one kind of item in the box, allow k-1 of them. Whenever you see a group of k different items show up (that is, there are k-1 types in the box, and the one arriving doesn't match any of them), you throw one of each type in the trash, including the one that just arrived. What data structure should we use for this "box"? Well, a bag, of course! As Misra and Gries explain, if the elements can be ordered, a tree-based bag with O(log k) basic operations will give the whole algorithm a complexity of O(n log k). One point to note is that the operation of removing one of each element is a bit expensive (O(k) for a typical implementation), but that cost is amortized over the arrivals of those elements, so it's no big deal. Of course, if your elements are hashable rather than orderable, you can use a hash-based bag instead, which under certain common assumptions will give even better asymptotic performance (but it's not guaranteed). If your elements are drawn from a small finite set, you can guarantee that. If they can only be compared for equality, then your bag gets much more expensive and I'm pretty sure you end up with something like O(nk) instead.
Find the majority element that appears n/2 times by Moore-Voting Algorithm
See method 3 of the given link for Moore's Voting Algo (http://www.geeksforgeeks.org/majority-element/).
Time:O(n)
Now after finding majority element, scan the array again and remove the majority element or make it -1.
Time:O(n)
Now apply Moore Voting Algorithm on the remaining elements of array (but ignore -1 now as it has already been included earlier). The new majority element appears n/4 times.
Time:O(n)
Total Time:O(n)
Extra Space:O(1)
You can do it for element appearing more than n/8,n/16,.... times
EDIT:
There may exist a case when there is no majority element in the array:
For e.g. if the input arrays is {3, 1, 2, 2, 1, 2, 3, 3} then the output should be [2, 3].
Given an array of of size n and a number k, find all elements that appear more than n/k times
See this link for the answer:
https://stackoverflow.com/a/24642388/3714537
References:
http://www.cs.utexas.edu/~moore/best-ideas/mjrty/
See this paper for a solution that uses constant memory and runs in linear time, which will find 3 candidates for elements that occur more than n/4 times. Note that if you assume that your data is given as a stream that you can only go through once, this is the best you can do -- you have to go through the stream one more time to test each of the 3 candidates to see if it occurs more than n/4 times in the stream. However, if you assume a priori that there are 3 elements that occur more than n/4 times then you only need to go through the stream once so you get a linear time online algorithm (only goes through the stream once) that only requires constant storage.
As you didnt mention space complexity , one possible solution is using hashtable for the elements which maps to count then you can just increment count if the element is found.
I have encountered an algorithmic problem but am not able to figure out anything better than brute force or reduce it to a better know problem. Any hints?
There are N bags of variable sizes and N types of items. Each type of items belongs to one bag. There are lots of items of each type and each item may be of a different size. Initially, these items are distributed across all the bags randomly. We have to place the items in their respective bags. However, we can only operate with a pair of bags at one time by exchanging items (as much as possible) and proceeding to the next pair. The aim is to reduce the total number of pairs. Edit: The aim is to find a sequence of transfers that minimizes the total number of bag pairs involved
Clarification:
The bags are not arbitrarily large (You can assume the bag and item sizes to be integers between 0 to 1000 if it helps). You'll frequently encounter scenarios where the all the items between 2 bags cannot be swapped due to the limited capacity of one of the bags. This is where the algorithm needs to make an optimisation. Perhaps, if another pair of bags were swapped first, the current swap can be done in one go. To illustrate this, let's consider Bags A, B and C and their items 1, 2, 3 respectively. The number in the brackets is the size.
A(10) : 3(8)
B(10): 1(2), 1(3)
C(10): 1(4)
The swap orders can be AB, AC, AB or AC, AB. The latter is optimal as the number of swaps is lesser.
Since I cannot come to an idea for an algorithm that will always find an optimal answer, and approximation of the fitness of the solution (amount of swaps) is also fine, I suggest a stochastic local search algorithm with pruning.
Given a random starting configuration, this algorithm considers all possible swaps, and makes a weighed decision based on chance: the better a swap is, the more likely it is chosen.
The value of a swap would be the sum of the value of the transaction of an item, which is zero if the item does not end up in it's belonging bag, and is positive if it does end up there. The value increases as the item's size increases (the idea behind this is that a larger block is hard to move many times in comparison to smaller blocks). This fitness function can be replaced by any other fitness function, it's efficiency is unknown until empirically shown.
Since any configuration can be the consequence of many preceding swaps, we keep track of which configurations we have seen before, along with a fitness (based on how many items are in their correct bag - this fitness is not related to the value of a swap) and the list of preceded swaps. If the fitness function for a configuration is the sum of the items that are in their correct bags, then the amount of items in the problem is the highest fitness (and therefor marks a configuration to be a solution).
A swap is not possible if:
Either of the affected bags is holding more than it's capacity after the potential swap.
The new swap brings you back to the last configuration you were in before the last swap you did (i.e. reversed swap).
When we identify potential swaps, we look into our list of previously seen configurations (use a hash function for O(1) lookup). Then we either set its preceded swaps to our preceded swaps (if our list is shorter than it's), or we set our preceded swaps to its list (if it's list is shorter than ours). We can do this because it does not matter which swaps we did, as long as the amount of swaps is as small as possible.
If there are no more possible swaps left in a configuration, it means you're stuck. Local search tells you 'reset' which you can do in may ways, for instance:
Reset to a previously seen state (maybe the best one you've seen so far?)
Reset to a new valid random solution
Note
Since the algorithm only allows you to do valid swaps, all constraints will be met for each configuration.
The algorithm does not guarantee to 'stop' out of the box, you can implement a maximum number of iterations (swaps)
The algorithm does not guarantee to find a correct solution, as it does it's best to find a better configuration each iteration. However, since a perfect solution (set of swaps) should look closely to an almost perfect solution, a human might be able to finish what the local search algorithm was not after it results in a invalid configuration (where not every item is in its correct bag).
The used fitness functions and strategies are very likely not the most efficient out there. You could look around to find better ones. A more efficient fitness function / strategy should result in a good solution faster (less iterations).
I want to create a doubly linked list with an order sequence (an integer attribute) such that sorting by the order sequence could create an array that would effectively be equivalent to the linked list.
given: a <-> b <-> c
a.index > b.index
b.index > c.index
This index would need to handle efficiently arbitrary numbers of inserts.
Is there a known algorithm for accomplishing this?
The problem is when the list gets large and the index sequence has become packed. In that situation the list has to be scanned to put slack back in.
I'm just not sure how this should be accomplished. Ideally there would be some sort of automatic balancing so that this borrowing is both fast and rare.
The naive solution of changing all the left or right indecies by 1 to make room for the insert is O(n).
I'd prefer to use integers, as I know numbers tend to get less reliable in floating point as they approach zero in most implementations.
This is one of my favorite problems. In the literature, it's called "online list labeling", or just "list labeling". There's a bit on it in wikipedia here: https://en.wikipedia.org/wiki/Order-maintenance_problem#List-labeling
Probably the simplest algorithm that will be practical for your purposes is the first one in here: https://www.cs.cmu.edu/~sleator/papers/maintaining-order.pdf.
It handles insertions in amortized O(log N) time, and to manage N items, you have to use integers that are big enough to hold N^2. 64-bit integers are sufficient in almost all practical cases.
What I wound up going for was a roll-my-own solution, because it looked like the algorithm wanted to have the entire list in memory before it would insert the next node. And that is no good.
My idea is to borrow some of the ideas for the algorithm. What I did was make Ids ints and sort orders longs. Then the algorithm is lazy, stuffing entries anywhere they'll fit. Once it runs out of space in some little clump somewhere it begins a scan up and down from the clump and tries to establish an even spacing such that if there are n items scanned they need to share n^2 padding between them.
In theory this will mean over time the list will be perfectly padded, and given that my IDs are ints and my sort orders are longs, there will never be a scenario where you will not be able to achieve n^2 padding. I can't speak to the upper bounds on the number of operations, but my guts tell me that by doing polynomial work at 1/polynomial frequency, that I'll be doing just fine.
Came across this interview question.
Write an algorithm to find the mean(average) of a large list. This
list could contain trillions or quadrillions of number. Each number is
manageable in hundreds, thousands or millions.
Googling it gave me all Median of Medians solutions. How should I approach this problem?
Is divide and conquer enough to deal with trillions of number?
How to deal with the list of the such a large size?
If the size of the list is computable, it's really just a matter of how much memory you have available, how long it's supposed to take and how simple the algorithm is supposed to be.
Basically, you can just add everything up and divide by the size.
If you don't have enough memory, dividing first might work (Note that you will probably lose some precision that way).
Another approach would be to recursively split the list into 2 halves and calculating the mean of the sublists' means. Your recursion termination condition is a list size of 1, in which case the mean is simply the only element of the list. If you encounter a list of odd size, make either the first or second sublist longer, this is pretty much arbitrary and doesn't even have to be consistent.
If, however, you list is so giant that its size can't be computed, there's no way to split it into 2 sublists. In that case, the recursive approach works pretty much the other way around. Instead of splitting into 2 lists with n/2 elements, you split into n/2 lists with 2 elements (or rather, calculate their mean immediately). So basically, you calculate the mean of elements 1 and 2, that becomes you new element 1. the mean of 3 and 4 is your new second element, and so on. Then apply the same algorithm to the new list until only 1 element remains. If you encounter a list of odd size, either add an element at the end or ignore the last one. If you add one, you should try to get as close as possible to your expected mean.
While this won't calculate the mean mathematically exactly, for lists of that size, it will be sufficiently close. This is pretty much a mean of means approach. You could also go the median of medians route, in which case you select the median of sublists recursively. The same principles apply, but you will generally want to get an odd number.
You could even combine the approaches and calculate the mean if your list is of even size and the median if it's of odd size. Doing this over many recursion steps will generate a pretty accurate result.
First of all, this is an interview question. The problem as stated would not arise in practice. Also, the question as stated here is imprecise. That is probably deliberate. (They want to see how you deal with solving an imprecisely specified problem.)
Write an algorithm to find the mean(average) of a large list.
The word "find" is rubbery. It could mean calculate (to some precision) or it could mean estimate.
The phrase "large list" is rubbery. If could mean a list or array data structure in memory, or the "list" could be the result of a database query, the contents of a file or files.
There is no mention of the hardware constraints on the system where this will be implemented.
So the first thing >>I<< would do would be to try to narrow the scope by asking some questions of the interviewer.
But assuming that you can't, then a complete answer would need to cover the following points:
The dataset probably won't fit in memory at the same time. (But if it does, then that is good.)
Calculating the average of N numbers is O(N) if you do it serially. For N this size, it could be an intractable problem.
An alternative is to split into sublists of equals size and calculate the averages, and the average of the averages. In theory, this gives you O(N/P) where P is the number of partitions. The parallelism could be implemented with multiple threads, with multiple processes on the same machine, or distributed.
In practice, the limiting factors are going to be computational, memory and/or I/O bandwidth. A parallel solution will be effective if you can address these limits. For example, you need to balance the problem of each "worker" having uncontended access to its "sublist" versus the problem of making copies of the data so that that can happen.
If the list is represented in a way that allows sampling, then you can estimate the average without looking at the entire dataset. In fact, this could be O(C) depending on how you sample. But there is a risk that your sample will be unrepresentative, and the average will be too inaccurate.
In all cases doing calculations, you need to guard against (integer) overflow and (floating point) rounding errors. Especially while calculating the sums.
It would be worthwhile discussing how you would solve this with a "big data" platform (e.g. Hadoop) and the limitations of that approach (e.g. time taken to load up the data ...)
Suppose you have several arrays of integers. What is a good way to find pairs of integers, not both from the same list, such that the difference between the first and second integer is 1?
Naturally I could write a naive algorithm that just looks through each other list until it finds such a number or hits one bigger. Is there a more elegant solution?
I only mention the condition that the difference be 1 because I'm guessing there might be some use to that knowledge to speed up the computation. I imagine that if the condition for a 'hit' were something else, the algorithm would work just the same.
Some background: I'm engaged in a bit of research mathematics and I seek to find examples of a certain construction. Any help would be much appreciated.
I'd start by sorting each array. Preferably with an algorithm that runs in O( n log(n) ) time.
When you've got a bunch of sorted arrays, you can set a pointer to the start of each array, check for any +/- 1 differences in the values of the pointers, and increment the value of the smallest-valued pointer, repeating until you've reached the max length of all but one of the arrays.
To further optimize, you could keep the pointers-values in a sorted linked list, and build the check function into an insertion sort. For each increment, you could remove the previous value from the list, and step through the list checking for +/- 1 comparison until you get to a term that is larger than a possible match. That way, if you're searching a bazillion arrays, you needn't check all bazillion pointer-values - you only need to check until you find a value that is too big, and ignore all larger values.
If you've got any more information about the arrays (such as the range of the terms or number of arrays), I can see how you could take advantage of that to make much faster algorithms for this through clever uses of array properties.
This sounds like a good candidate for the classic merge sort where the final stage is not a unification but comparison.
And the magnitude of the difference wouldn't affect this, but thanks for adding the information.
Even though you state the second list is in an array, if you could put it in a hashmap of some sort then you could make it faster than just the naive approach.
Basically,
Loop through the first array.
Look to see if there exists an object in the hashmap that is one larger than the current array value.
That way you can build up pairs of numbers that meet your requirements.
I don't know if it would be as flexible as you would like though.
Basically, you may want to consider other data structures, to help you find a better solution.
You have o(n log n) from the sorting.
You can also the the search in o(log n) for each element, if you have some dynamic queryset. You can sort the arrays and then for each element in the first array binary search his upper_bound and lower_bound in the second array and check the difference.